Solution to boundary shape identification problems in elliptic boundary value problems using shape derivatives

Abstract

This chapter briefly defines the identification problems of geometrical boundary shapes of domains in which elliptic boundary value problems are treated. These problems are formulated as minimization problems of squared error integrals between the actual solutions of the elliptic boundary value problems and its reference data with respect to perturbation of the uncertain boundary. The fundamental theory concerning with the shape derivatives of functional with respect to domain perturbation and the gradient method in Hilbert space are also presented by mathematicians. Based on the theories, this chapter provides a concrete solution to the geometrical domain identification problems. It briefly describes the derivation of the shape gradient functions for the shape identification problems of two types referring to boundary value on sub boundary and referring to gradient in subdomain, and introduces the definition of the gradient method in Hilbert space.